On a reformulation of the commutator subgroup

TL;DR

This paper explores the relationship between a specific congruence on groups and the classical commutator subgroup, showing they coincide and relate to the abelianization of the group.

Contribution

It establishes that for groups, the orientable subsemigroup corresponds exactly to the commutator subgroup, linking a previous semigroup concept to fundamental group theory.

Findings

Orientable(G) equals the commutator subgroup [G,G]

The quotient G / σ_orient is the abelianization of G

Provides a new perspective on the structure of groups via congruences

Abstract

For semigroup $S$, a commutative congruence $\sigma_{orient}$ on $S$ and a subsemigroup Orientable($S$) of $S$ were introduced in "Two cancellative commutative congruences and group diagrams", Semigroup Forum (2011) 82: 338-353. Here we demonstrate that when the semigroup is in fact a group $G$, then Orientable($G$) is the commutator subgroup $[G,G]$ and $ G / \sigma_{orient}$ is the abelian quotient group $G / [G,G]$.